For example, your app icon would be blue and black and your logo on printed shirts would be red and black.
I suppose the Yes/No answer would have to be a "No", however, there are some considerations.
The purpose of a logo is to build recognition by repetition. Therefore, diluting the logo with variations (be they in shape or in color) are likely to decrease its effectiveness since the consumer would take a little longer to reach the point of instant recognition.
Having said that, if you feel strongly about having logo variations (perhaps to represent different products that you're offering?), try keeping a core of the logo unchanged and build variations only on certain elements. The most notable example that I can think of is FedEx: see http://i.stack.imgur.com/GGATc.png.
You mentioned printed shirts so that brought up the topic of branded merchandise. If your brand will be present on a variety of merchandise, keep in mind that many colors won't look good against certain backgrounds and textures. If you must have / offer merchandise that comes in both very light and very dark backgrounds, it'd be a good idea to let your designer know that before creating and settling for a color scheme.